Let $T:X\to X$ be a linear operator on a Banach space $X$. We know that the spectrum of $T$ is an upper semicontinuous function of $T$ for the uniform convergence: that is, if $T_n:X\to X$ is a sequence of operators, s.t, $T_n\to T$ uniformly then, given an open set $G$ containing the spectrum of $T$, $\sigma(T)$, there must exist $k$ s.t $n>k$ implies that $\sigma (T_n)$ is contained in $G$.
I would like to know if this result remains valid under weaker conditions, like $T_n$ converges to $T$ in the compact parts of $X$.
I'd like to know if properties like spectral gap in
the spectrum remain if we weaken the notion of